This paper examines the Foldy—Wouthuysen and Feynman—Gell-Mann representations of the Dirac equation. The analysis is conducted for electrons and positrons interacting with electromagnetic fields. Versions of quantum electrodynamics are considered both within the scope of perturbation theory and in the nonperturbative case with strong electromagnetic fields. Mathematical artifacts that contradict the physical premises of the theory are identified in the studied representations of the Dirac equation. These mathematical paradoxes are resolved if the theory only employs amplitude states (real and virtual) with positive energies.

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Keywords: Dirac equation representations, quantum electrodynamics, fermion vacuum, positive and negative energy states, mathematical paradoxes of the theory PACS: 03.65.Pm, 11.10.St, 12.20.−m (all) DOI: 10.3367/UFNe.2025.11.040057 URL: https://ufn.ru/en/articles/2025/12/h/ Citation: Neznamov V P "Mathematical paradoxes of Dirac equation representations" Phys. Usp. 68 (12) (2025) Received: 2nd, June 2025, revised: 6th, November 2025, accepted: 11th, November 2025 Оригинал: Незнамов В П «Математические парадоксы представлений уравнения Дирака» УФН 195 1356–1361 (2025); DOI: 10.3367/UFNr.2025.11.040057